This invention relates to interferometry, and more particularly to phase-shifting interferometry.
Interferometric optical techniques are widely used to measure optical thickness, flatness, and other geometric and refractive index properties of precision optical components such as glass substrates used in lithographic photomasks.
For example, to measure the surface profile of a measurement surface, one can use an interferometer to combine a measurement wavefront reflected from the measurement surface with a reference wavefront reflected from a reference surface to form an optical interference pattern. Spatial variations in the intensity profile of the optical interference pattern correspond to phase differences between the combined measurement and reference wavefronts caused by variations in the profile of the measurement surface relative to the reference surface. Phase-shifting interferometry (PSI) can be used to accurately determine the phase differences and the corresponding profile of the measurement surface.
With PSI, the optical interference pattern is recorded for each of multiple phase-shifts between the reference and measurement wavefronts to produce a series of optical interference patterns that span at least a full cycle of optical interference (e.g., from constructive, to destructive, and back to constructive interference). The optical interference patterns define a series of intensity values for each spatial location of the pattern, wherein each series of intensity values has a sinusoidal dependence on the phase-shifts with a phase-offset equal to the phase difference between the combined measurement and reference wavefronts for that spatial location. Using numerical techniques known in the art, the phase-offset for each spatial location is extracted from the sinusoidal dependence of the intensity values to provide a profile of the measurement surface relative the reference surface. Such numerical techniques are generally referred to as phase-shifting algorithms.
The phase-shifts in PSI can be produced by changing the optical path length from the measurement surface to the interferometer relative to the optical path length from the reference surface to the interferometer. For example, the reference surface can be moved relative to the measurement surface. Alternatively, the phase-shifts can be introduced for a constant, non-zero optical path difference by changing the wavelength of the measurement and reference wavefronts. The latter application is known as wavelength tuning PSI and is described, e.g., in U.S. Pat. No. 4,594,003 to G. E. Sommargren.
PSI data can be analyzed using PSI algorithms. A PSI algorithm presumes a certain phase shift (e.g., 45° or 90°, depending on the algorithm) between successive intensity values acquired at each spatial location of the interference pattern during the phase shifting. In other words, a PSI algorithm assumes that the intensity pattern will vary at a particular frequency which corresponds to a set phase shift between each intensity value. The PSI algorithm extracts a phase of the intensity pattern at each spatial location based on the phase shift. Because each PSI algorithm assumes a certain phase shift in extracting phase information from the interference pattern, any deviation from presumed phase shift will compromise the accuracy of the extracted phases, reducing the fidelity of a measurement. Examples of PSI algorithms are described, for example, in U.S. patent application Ser. No. 09/349,593, entitled “METHOD AND SYSTEM FOR PROFILING OBJECTS HAVING MULTIPLE REFLECTIVE SURFACES USING WAVELENGTH-TUNING,” filed on Jul. 9, 1999, to Peter de Groot.
In order to ensure that the interference occurs with a frequency appropriate for a PSI algorithm (i.e., the phase shift between intensity values is appropriate for the algorithm), the rate at which the optical frequency varies during wavelength tuning should be set to an appropriate value. The interference frequency, f, is a function of both the tuning rate, {dot over (ν)}, and the cavity total optical path length (OPL), D, via                     f        =                                                            v                .                            ⁢                                                           ⁢              D                        c                    ,                                    (        1        )            where c is the speed of light. Thus, according to equation 1, the cavity OPL should be known in order to set a tuning rate at a value such that the interference occurs with a frequency appropriate for the PSI algorithm being used. Typically, the user will input a nominal value for D, from which a suitable tuning rate can be set.
More recently, alternative techniques for analyzing PSI data have been disclosed in which PSI data is transformed into a domain that produces spectrally separated peaks each corresponding to a particular pair of surfaces in an interferometric cavity defined by multiple pairs of surfaces. Each peak provides optical path length and surface reflectivity information about a corresponding pair of surfaces in the cavity. As a result, the interferometric data from such cavities provides simultaneous information about multiple surfaces. For example, information about any particular surface may be determined generically. Such PSI analysis techniques, referred to as Frequency Transform Phase Shifting Interferometry (“FTPSI”), are described in, for example, U.S. patent application Ser. No. 09/919,511, entitled “FREQUENCY TRANSFORM PHASE SHIFTING INTERFEROMETRY,” filed on Jul. 31, 2001
For cavities formed by a single pair of surfaces, this latter technique does not require prior knowledge of the cavity OPL in order to set an appropriate tuning rate, because the transform provides information for a range of frequencies. Information may be determined without the need for a preset phase interval between adjacent data points, the phase interval being associated with a particular cavity length of interest. The interferometric phase of each cavity can be determined from the transform of the interference data, evaluated approximately at the peak frequency.
However, in these embodiments, the frequency range of the transform should be appropriate for the interference frequency of the cavity. In particular, the optical frequency tuning rate should being sufficiently high to resolve the cavity of interest. For example, in the case of FTPSI utilizing a Fourier transform, the spectral resolution limit, fmin, is inversely proportional to the observation time, Δt, and the minimal resolvable interference frequency is                               f          min                =                                            1              +              μ                                      Δ              ⁢                                                           ⁢              t                                =                                                                      (                                      1                    +                    μ                                    )                                ⁢                                  f                  S                                            N                        ⁢                                                   .                                              (        2        )            First order frequencies should be separated by at least fmin to be resolved. In equation 2, fs is the sampling frequency, N is the total number of intensity samples acquired, and the parameter μ is introduced as a practical matter. The theoretical resolution limit occurs when μ=0, but in practice, the minimum resolvable frequency should be somewhat larger to account for potential instrumental deficiencies and phase error sensitivities. Setting f in equation 1 to fmin, the minimum resolvable OPL, Dmin, can be expressed as                               D          min                =                                                            c                ⁡                                  (                                      1                    +                    μ                                    )                                            ⁢                              f                S                                                                    v                .                            ⁢                                                           ⁢              N                                ⁢                                           .                                    (        3        )            In other words, the spectral resolution is inversely proportional to the tuning rate. Hence, a relatively small tuning rate can result in poor resolution (i.e., a relatively large resolution limit).
In addition, for a fixed sampling frequency, the tuning rate should be low enough to avoid aliasing. Aliasing occurs due to the number of samples acquired during a tune being finite. Aliasing can occur when the sampling frequency is less than or equal to the Nyquist frequency (also known as the folding frequency). Accordingly, in order for the frequency, f, associated with a cavity to be detected without aliasing, the sampling frequency should be more than twice f. Thus, when the tuning rate is too high, the contribution to the interference signal at f can be aliased to a lower frequency, potentially corrupting the data.
Therefore, even when using FTSPI, it is desirable to precalibrate the interferometry system by selecting a tuning rate appropriate for the cavity OPL using methods disclosed herein.